for some weighting parameter \(\alpha \in \left[0,1\right]\) . Both \(\mathcal {T}_1\)  and \(\mathcal {T}_2\)  are based on the maximum mean discrepancy (MMD) to measure the empirical distance between the marginal distributions [1]}. Let \(k: \mathcal {D}\times \mathcal {D} \rightarrow \mathbb {R}^{+}\)  be a symmetric definite positive kernel. The existence of a dot product space \(\mathcal {H}\)  and feature mapping \(\phi : \mathcal {D} \rightarrow \mathcal {H}\)  is guaranteed, such that \(k(X,X^{\prime })=\langle \phi (X),\phi (X^{\prime })\rangle _{\mathcal {H}}\) . A reproducing kernel of \(\mathcal {H}\)  is a kernel function that satisfies (1) \(\forall X\in \mathcal {D}\) , \(k(\cdot ,X)\in \mathcal {H}\) , and (2) \(\forall X\in \mathcal {D}\) , \(\forall g\in \mathcal {H}\) , \(\langle g,k(\cdot ,X) \rangle _{\mathcal {H}}=g(X)\) . \(\mathcal {H}\)  is then said to be a reproducing kernel Hilbert space (RKHS). Kernel mean embedding results from extending the mapping \(\phi \)  to the space of probability distributions by representing each distribution as a mean function \(\phi (F) = \mathbf {E}[k(\cdot ,X)] = \int _{\mathcal {D}} k(\cdot ,X)dP\) . The kernel mean embedding can be empirically estimated by \(\widetilde{\phi }= \frac{1}{n}\sum _{i=1}^{n}k\left(\cdot , X\right)\) . Then, we can measure the distance between random samples as follows:
\(\mathcal {T}_1\left(\mathbf {X}^{\text{com}}_1,\mathbf {X}^{\text{com}}_2\right)&= \left\Vert \widetilde{\phi }^{\text{com}}_{1}-\widetilde{\phi }^{\text{com}}_{2}\right\Vert _{\mathcal {H}}^{2}\\&= \langle \frac{1}{n_1}\sum _{i=1}^{n_1}k\left(\cdot , X_{1i}\right)-\frac{1}{n_1}\sum _{i=1}^{n_1}k\left(\cdot , X_{2i}\right), \frac{1}{n_1}\sum _{i=1}^{n_1}k\left(\cdot , X_{1i}\right)-\frac{1}{n_1}\sum _{i=1}^{n_1}k\left(\cdot , X_{2i}\right) \rangle \\&=\frac{1}{n_1^{2}} \sum _{i=1}^{n_1}\sum _{j=1}^{n_1} k\left(X_{1i},X_{1j}\right)+\frac{1}{n_1^{2}} \sum _{i=1}^{n_1}\sum _{j=1}^{n_1} k\left(X_{2i},X_{2j}\right)-\frac{2}{n_1^{2}} \sum _{i=1}^{n_1}\sum _{j=1}^{n_1}k\left(X_{1i},X_{2j}\right).\) 
